#### Problem 53E

53. Show that the ellipse $x^{2} / a^{2}+y^{2} / b^{2}=1$ and the hyperbola $x^{2} / A^{2}-y^{2} / B^{2}=1$ are orthogonal trajectories if $A^{2}<a^{2}$ and $a^{2}-b^{2}=A^{2}+B^{2}$ (so the ellipse and hyperbola have the same foci).